Bouwmeester and Tanguay scored 89 seconds apart in the third quarter to erase Edmonton's early lead in the first match-up of the young season between the provincial rivals.
Miikka Kiprusoff stopped 24 shots for the Flames, who had lost three of their first four games of the season.
Devan Dubnyk turned in 26 saves for the Oilers, who held Calgary to 13 shots on goal in the first two periods but allowed 12 in the third. Edmonton finished with 25 shots on goal.
Tom Gilbert gave Edmonton a 1-0 lead on a power-play goal at 6:42 of the first period.
Bouwmeester tied the contest on a wrist shot at 14:03 of the third period, and Tanguay scored the game-winner with a wrister at 15:32.